Graphing a set of Complex Numbers

Question

I came across the following question:

Make a graph of the following set:

$$E=\{z \in \mathbb{C}\big| |z+i|=2|z|\} $$

But I have no clue how to find elements from this set. I looked at some more trivial numbers, such as $0$,$1$ and $i$ but non seem to be in this set. Also simplifying the condition doesn't give me any clue.

Can someone please help me with understanding this set?

Answer 1

The complex numbers in question correspond to certain points in $R^2$ as shown below

So, the question can be formulated as a problem in coordinate geometry: Picture the curve described by the following equation:

$$\sqrt{(y+1)^2+x^2}=2\sqrt{(x^2+y^2)}$$

or $$(y+1)^2+x^2=4(x^2+y^2).$$

We get then

$$3y^2-2y+3x^2-1=0$$

which is the equation of a circle of radius $\frac23$ centered at $\left(0,\frac13\right)$.

Indeed. Divide both sides of the equation above by $3$ and rearrange again:

$$y^2-\frac23y+x^2-\frac13=0.$$

Taking the complete square of the $y$ term we get

$$\left(y-\frac13\right)^2+x^2=\frac49.$$

Answer 2

I'd rewrite the condition as $\left|\frac{z-i}{z}\right|=2$, then get rid of tge modulus: $$ \frac{z-i}{z}=2e^{it} $$ Now solve for $z$. Then plug in various $t$ and see what $z$ you get.

Answer 3

When $\text{z}\in\mathbb{C}$:

• $$\left|\text{z}+i\right|=\left|\left(\Re\left[\text{z}\right]+\Im\left[\text{z}\right]i\right)+i\right|=\left|\Re\left[\text{z}\right]+\left(1+\Im\left[\text{z}\right]\right)i\right|=\sqrt{\Re^2\left[\text{z}\right]+\left(1+\Im\left[\text{z}\right]\right)^2}$$
• $$2\left|\text{z}\right|=2\left|\Re\left[\text{z}\right]+\Im\left[\text{z}\right]i\right|=2\sqrt{\Re^2\left[\text{z}\right]+\Im^2\left[\text{z}\right]}$$

So, when set those two equal:

$$\sqrt{\Re^2\left[\text{z}\right]+\left(1+\Im\left[\text{z}\right]\right)^2}=2\sqrt{\Re^2\left[\text{z}\right]+\Im^2\left[\text{z}\right]}$$